Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
The authors obtained some geometric results on certain new classes of analytic
functions involving sigmoid function defined by Fadipe-Joseph et. al. 2016 as
Tγ(λ, β, α, µ). Extreme point property, radius of starlikeness and convexity, con-
volution property and Fekete-Szego inequality for the class were proved.
Results & Lemmas (8)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 2.1
Lemma 2.1 [1] Let fγ(z) ∈Tγ defined as fγ(z) = z −P∞ k=2 γ(s)akzk (ak ≥0 and γ ̸= 0). If fγ ∈Tγ(λ, β, α, µ), then ∞ X k=2 γ(s)kn γ(s)k[1…
Lemma 2.1 [1] Let fγ(z) ∈Tγ defined as fγ(z) = z −P∞ k=2 γ(s)akzk (ak ≥0 and γ ̸= 0). If fγ ∈Tγ(λ, β, α, µ), then ∞ X k=2 γ(s)kn γ(s)k[1 −β(1 −2α)] −β λ + 2αµ + 1 β |ak|
Theorem 3.1
Theorem 3.1 Extreme points for class Tγ(λ, β, α, µ). If a function fγ(z) defined by (1.1) belongs to the class Tγ(λ, β, α, µ). Let f1(z) =…
Theorem 3.1 Extreme points for class Tγ(λ, β, α, µ). If a function fγ(z) defined by (1.1) belongs to the class Tγ(λ, β, α, µ). Let f1(z) = z and fγ(z) = γ(s)(1 −β) + β[2α(γ(s) −µ) −λ] −µ γ(s)kn h γ(s)k[1 −β(1 −2α)] −β λ + 2αµ + 1 β izk, k ≥2. The function fγ ∈Tγ(λ, β, α, µ) if and only if it can be expressed in the form fγ(z) = ∞
Theorem 3.2
Theorem 3.2 Fekete-Szegˇo inequality for class Tγ(λ, β, α, µ). If a function f(z) ∈Tγ defined by (1.1) belongs to the class Tγ(λ, β, α, µ)…
Theorem 3.2 Fekete-Szegˇo inequality for class Tγ(λ, β, α, µ). If a function f(z) ∈Tγ defined by (1.1) belongs to the class Tγ(λ, β, α, µ) and µ ∈R. Then, |a3 −σa2 2| ≤ B3[B2 2 −σB3B1] B1B2 2 .
Theorem 3.3
Theorem 3.3 (Starlikeness): Let the function fγ(z) defined by (1.1) be in the class Tγ(λ, β, α, µ); then fγ(z) is starlike of order σ (0 ≤δ…
Theorem 3.3 (Starlikeness): Let the function fγ(z) defined by (1.1) be in the class Tγ(λ, β, α, µ); then fγ(z) is starlike of order σ (0 ≤δ < 1) in |z| < r1, where r1 = infk (1 −δ)γ(s)kn h γ(s)k[1 −β(1 −2α)] −β λ + 2αµ + 1 β i [γ(s)(1 −β) + β[2α(γ(s) −µ) −λ] −µ]γ(s)(k −δ)
Theorem 3.4
Theorem 3.4 (Convexity): Let the function fγ(z) defined by (1.1) be in the class Tγ(λ, β, α, µ); then fγ(z) is convex of order δ (0 ≤δ < 1)…
Theorem 3.4 (Convexity): Let the function fγ(z) defined by (1.1) be in the class Tγ(λ, β, α, µ); then fγ(z) is convex of order δ (0 ≤δ < 1) in |z| < r2, where r2 = infk (1 −δ)γ(s)kn h γ(s)k[1 −β(1 −2α)] −β λ + 2αµ + 1 β i k(k −δ)γ(s)[γ(s)(1 −β) + β[2α(γ(s) −µ) −λ] −µ]
Theorem 3.5
Theorem 3.5 (Close-to-convex): Let the function fγ(z) defined by (1.1) be in the class Tγ(λ, β, α, µ). Then fγ(z) is closed-to-convex of…
Theorem 3.5 (Close-to-convex): Let the function fγ(z) defined by (1.1) be in the class Tγ(λ, β, α, µ). Then fγ(z) is closed-to-convex of order δ (0 ≤δ < 1) in |z| < r3, where r3 ≤ (1 −δ)γ(s)kn h γ(s)k[1 −β(1 −2α)] −β λ + 2αµ + 1 β i kγ(s)[γ(s)(1 −β) + β[2α(γ(s) −µ) −λ] −µ]
Theorem 3.6
Theorem 3.6 Neighbourhood Property for class Tγ(λ, β, α, µ). Let ς = γ(s)(1 −β) + β[2α(γ(s) −µ) −λ] −µ γ(s)kn h γ(s)k[1 −β(1 −2α)] −β λ +…
Theorem 3.6 Neighbourhood Property for class Tγ(λ, β, α, µ). Let ς = γ(s)(1 −β) + β[2α(γ(s) −µ) −λ] −µ γ(s)kn h γ(s)k[1 −β(1 −2α)] −β λ + 2αµ + 1 β i (3.7) Then Tγ(λ, β, α, µ) ⊂N(n,ς)e(z).
Theorem 3.7
Theorem 3.7 Convolution Property for class Tγ(λ, β, α, µ). Let fγ(z) and gγ(z) defined by (1.1) and (1.2)) respectively be members of Tγ(λ,…
Theorem 3.7 Convolution Property for class Tγ(λ, β, α, µ). Let fγ(z) and gγ(z) defined by (1.1) and (1.2)) respectively be members of Tγ(λ, β, α, µ) such that hγ(z) = z −P∞ k=2 γ(s)akbkzk where akbk ≥0 as defined by (1.4). Then, hγ(z) is in the subclass of Tγ(λ, β, α, µ) where γ(s)(1 −β) + β[2α(γ(s) −µ) −λ] −µ1 γ(s)kn h γ(s)k[1 −β(1 −2α)] −β λ + 2αµ + 1 β i ≤γ(s)(1 −β) + β[2α(γ(s) −µ) −λ] −µ1 γ(s)(1 −β) + β[2α(γ(s) −µ) −λ] −µ2
Definitions (1)
Def 2.1
Definition 2.1 Sˇalˇagean Differential Operator Involving Modified Sigmoid Function Dnfγ(z) = γn(s)z + ∞ X k=2 γm(s)knakzk; m = n + 1 (2.1)…
Definition 2.1 Sˇalˇagean Differential Operator Involving Modified Sigmoid Function Dnfγ(z) = γn(s)z + ∞ X k=2 γm(s)knakzk; m = n + 1 (2.1) for details see [1]. Definition 2.2 A function fγ ∈Tγ defined by (1.1) belongs to the class Tγ(λ, β, α, µ) if
Function classes studied:
Related Papers